Calculus Examples

$y = \frac{1}{2 \sqrt{x}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$y = \frac{1}{2 x^{\frac{1}{2}}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{1}{2 x^{\frac{1}{2}}})$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{1}{2 x^{\frac{1}{2}}}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]$ .

$\frac{1}{2} \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]$

Step 4.2

Apply basic rules of exponents.

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Step 4.2.1

Rewrite $\frac{1}{x^{\frac{1}{2}}}$ as ${(x^{\frac{1}{2}})}^{- 1}$ .

$\frac{1}{2} \frac{d}{d x} [{(x^{\frac{1}{2}})}^{- 1}]$

Step 4.2.2

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 1}$ .

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Step 4.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{2} \frac{d}{d x} [x^{\frac{1}{2} \cdot - 1}]$

Step 4.2.2.2

Combine $\frac{1}{2}$ and $- 1$ .

$\frac{1}{2} \frac{d}{d x} [x^{\frac{- 1}{2}}]$

Step 4.2.2.3

Move the negative in front of the fraction.

$\frac{1}{2} \frac{d}{d x} [x^{- \frac{1}{2}}]$

Step 4.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - \frac{1}{2}$ .

$\frac{1}{2} (- \frac{1}{2} x^{- \frac{1}{2} - 1})$

Step 4.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} (- \frac{1}{2} x^{- \frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 4.5

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{1}{2} (- \frac{1}{2} x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 4.6

Combine the numerators over the common denominator.

$\frac{1}{2} (- \frac{1}{2} x^{\frac{- 1 - 1 \cdot 2}{2}})$

Step 4.7

Simplify the numerator.

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Step 4.7.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} (- \frac{1}{2} x^{\frac{- 1 - 2}{2}})$

Step 4.7.2

Subtract $2$ from $- 1$ .

$\frac{1}{2} (- \frac{1}{2} x^{\frac{- 3}{2}})$

Step 4.8

Move the negative in front of the fraction.

$\frac{1}{2} (- \frac{1}{2} x^{- \frac{3}{2}})$

Step 4.9

Combine $x^{- \frac{3}{2}}$ and $\frac{1}{2}$ .

$\frac{1}{2} (- \frac{x^{- \frac{3}{2}}}{2})$

Step 4.10

Multiply $\frac{1}{2}$ by $\frac{x^{- \frac{3}{2}}}{2}$ .

$- \frac{x^{- \frac{3}{2}}}{2 \cdot 2}$

Step 4.11

Simplify the expression.

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Step 4.11.1

Multiply $2$ by $2$ .

$- \frac{x^{- \frac{3}{2}}}{4}$

Step 4.11.2

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{4 x^{\frac{3}{2}}}$

Step 5

Reform the equation by setting the left side equal to the right side.

$y' = - \frac{1}{4 x^{\frac{3}{2}}}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{1}{4 x^{\frac{3}{2}}}$